Interestingly, ED therefore has a problem that is somewhat inverted: how does one construct a Lagrangian formulation given a Hamiltonian? This, it seems is, in fact, not straightforward, or potentially, not possible at all. For example, Reginatto and Hall [1]} suggest that a Lagrangian formulation for the variables \(\rho (x)\)  and \(\Phi (x)\)  would require the introduction of two additional variables to play the role of “velocities" corresponding to the “coordinates" \(\rho (x)\)  and \(\Phi (x)\) .
